I was listening to the radio the other day when they posed an interesting question. A certain type of car has 47 owners in the UK. There are 66 million people in the UK and their radio show has 2.1 million listeners. What is the chance that one of those car owners is listening in?

We have to make a few assumptions. Firstly, as they quickly pointed out, not all of those listeners are in the UK thanks to listening over the internet, but we'll ignore that. Secondly we have no data whatsoever about a correlation between people who own this make of car and people who listen to the radio so we'll assume they are completely independent. Thirdly, they presumably actually cared about whether someone was listening at that moment and could/would call in. We don't. (I didn't get to listen to the full show so have no idea whether anyone called in or not.)

Dave and I propose that the best way to work this out is actually 1 - (probability that none of the car owners are listeners). If the total population is size $n$, there are $j$ car owners and $k$ listeners, then this should be:

$$1 - ((n - k)/n * (n - k - 1)/(n - 1) * ...)$$ with $j$ terms in the brackets. Or more formally: $$1 - Π↙{i=0}↖{j-1} ({n - k - i}/{n - i})$$

Here's a calculator for this formula:

Total population:
Car owners:

And to check it, a simulator to test against. Note that it's inefficient so can't run the original problem, but try out smaller population sizes to see that the results are in line.

Number of simulations: